copy and connect a part of an object
Work on model's topology

Freestyle takes also in account the topological features of each edge to determine whether or not it belongs to one of the Line set category (and so if it should be drawn).

You must ensure to provide to the software a model that could be interpreded the way you want.



Starting from a similar situation:



There are several different ways you can follow to model a similar object. In this case, for example, I choose to make room for the duplicating geometry by deleting the faces, duplicating the geometry and snapping it to the holeìs border.

 

Now the geometry is in-place, but is not connected. Remove doubles operator is capable of joining the vertices (so relateded edges too) withing a given threshold.



Results:

 



Another possible solution for this specific situation could be to leave intact the stick and just overlay a disconnected copy of the geometry.

You must avoid overlapping faces (overlapping edges shouldn't represent a big issue in this case) so you should duplicate only the selection shown in the images beleow:

 



Note: all "default" Frestyle line set style settings had been used in this demonstration apart from the Crease angle that hase been increased to 150°.